Use mathematical induction to show that the product of any $n$ consecutive positive integers is divisible by $n!.$

Hint: Use the identity

$m(m+1)cdots\frac{m+n-1}{n}(m-1)m(m+1)cdots\frac{m+n-2}{n}!+m(m+1)cdots\frac{m+n-2}{n-1}!$

Design a Turing Machine that given two $16-$bit binary numbers written on the tape, delimited by A$,$ computes

the sum of both numbers. For example, if the tape initially contains:

$0001001001100111A0001000100100001$Axxxxxxxxxxxxxxxx

The final tape, when the Turing Machine stops $($enters its final state$),$ must contain:

XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXA$0010001110001000$A